Axis of rotation, plane of reflection

• JamesGoh
In summary, for a 3x3 orthogonal matrix with determinant=-1, the axis of rotation is the same as the plane of reflection. This is because the matrix can be written as a product of two unitary matrices, one of which has only real entries, resulting in a rotation and reflection around the same axis. However, it is necessary to prove that this matrix is proportional to a real matrix.
JamesGoh
For a 3x3 orthogonal matrix with determinant= -1 (which means rotation followed by simple reflection), is the axis of rotation the same as the plane of reflection ?

My reasoning is follows (see attachment)

Say you have two vectors with the same angle size (which i call A), same x-values, but one of the z-values (height in this case), is the negative of the other (n.b. y-value is zero in both vectors )

Vector 1 rotates around the x-axis by 2A to get to the same spot as vector 2. Because the angle size is the same and because the z-component of vector 2 is the negative of the z-component of vector 1, we get a reflection ?

Since the "reflection" happens about the x-axis, this is why the plane of reflection is the same as the axis of rotation in the case of a 3x3 orthogonal matrix having determinant 1 ?

Attachments

• examplepicture.jpg
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if $A\in \textrm{O}(3,\mathbb{R})$ and $\det(A)=-1$, there exists a $U\in \textrm{SU}(3)$ such that

$$UAU^{\dagger} = \left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & e^{i\theta} & 0 \\ 0 & 0 & e^{-i\theta} \\ \end{array}\right)$$

with some $\theta\in\mathbb{R}$.

Then there exists a $V\in \textrm{SU}(2)$ such that

$$V\left(\begin{array}{cc} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \\ \end{array}\right)V^{\dagger} = \left(\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array}\right)$$

So if you define

$$W = \left(\begin{array}{cc} 1 & 0 \\ 0 & V \\ \end{array}\right)U$$

then $WAW^{\dagger}$ will be of such form that reflection and rotation are clearly carried out with respect to the same axis. Only problem is that $W$ doesn't necessarily have only real entries. How to prove that $W$ is necessarily proportional to a real matrix?